Chapter 4

PID Pole-Placement Controller

4.1 Introduction

For many years ago a great deal of attention has

been paid to the problem of designing pole-placement

controller. Various linear control designs based on

classical pole-placement ideas were developed and

employed in real application. The pole placement has

proved to be one of the most successful design

methods for linear control systems. Ultimately, each

design specification leads, either directly or

indirectly, to a particular assignment of the poles

of the system. The practical designs, however,

rarely call for exact pole positions [8]. In the

practical designs, the PID controllers are widely

used. This may be attributed to the fact that PID

controllers has simple structure, and easy to

maintained and tune. Therefore it is desirable to

combine the advantages of pole-placement and PID

control.

4.2 PID Controller

The PID controller is the most common form of

feedback control systems. It was an essential

23

Chapter 4

element of early governors and it became the

standard tool when process control emerged in the

1940s.

In process control today, more than 95% of the

control loops are of PID type, most loops are

actually PI control. PID controllers are today found

in all areas where control is used [9].

The controllers come in many different forms. There

are standalone systems in boxes for one or a few

loops, which are manufactured by the hundred

thousands yearly. PID control is an important

ingredient of a distributed control system. The

controllers are also embedded in many special

purposes

control systems. PID control is often combined with

logic, sequential functions, selectors, and simple

function blocks to build the complicated automation

systems used for energy production, transportation,

and manufacturing. Many sophisticated control

strategies, such as model predictive control, are

also organized hierarchically. PID control is used

at the lowest level, the multivariable controller

gives the set points to the controllers at the lower

24

Chapter 4

level. The PID controller can thus be said to be an

important component in every control engineer’s tool

box.

PID controllers have survived many changes in

technology, from mechanics and pneumatics to

microprocessors via electronic tubes, transistors,

integrated circuits. The microprocessor has had a

dramatic influence on the PID controller.

Practically all PID controllers made today are based

on microprocessors.

This has given opportunities to provide additional

features like automatic tuning, gain scheduling, and

continuous adaptation [10].

4.3 Conventional PID ControllerPID controller is a one of the earliest industrial

controllers. It has many advantages: Its cost is

economic , simple and easy to be tuned and

robust .

This controller has been proven to be remarkably

effective in regulating a wide range of processes.

It does not require an exact model and hence, it can

25

Chapter 4

be used for processes whose models are considerably

difficult to be driven.

A good survey is found in [11].

In general there are two approaches in PID tuning:

- Model based approach, if the process model is

available.

- Non-model based approach, if the process model is

difficult to be driven.

In the second approach, Ziegler and Nichols method

can be applied based on used the relay feedback to

estimate the limit cycle and then tune the PID

parameters.

However, in spit of the advantages of the PID

controller, there remain several drawbacks. It

cannot cope well in some cases such as:

- Non-linear processes, (changing in operating

point).

- Time-varying parameters.

- Compensation of strong and rapid disturbances.

- Supervision in multivariable control.

PID controller is simple and linear; it can give a

good performance for stable

26

Chapter 4

linear processes. Self-tuning and adaptive PID

design approaches can overcome the operating point

varying parameters. However, this requires a high

capacity of computations and makes the PID

performance not guaranteed.

Figure (4.1.a) General structure of continuous time

PID controller

PID controller consists of three terms:

- Proportional action.

- Derivative action to speed up the response.

- Integral action to eliminate the steady state

error.

Combining all the previous three control action, we

have the conventional PID controller which finds

extensive application in industrial control. For the

pksTk p )(

GAActuator

Gp process+-

_

_

--

++

)(tu

Controller

27

Chapter 4

continuous time case, the controller in its basic

from is described by the integral-differential

equation [10].

(4.1)

The transfer function is:

PID controller: (4.2)

: Proportional gain

: Derivative time

: Integral time

And also:

Figure (4.1.a) presents a block diagram of three

term controller . In the case of discrete time

system, the PID controller can be described in its

simplest form by the following difference equation.

(4.3)

Where T is sampling time.

After some algebraic manipulations, may be

written as:

28

Chapter 4

(4.4)

Where

(4.5)

(4.6)

(4.7)

Figure (4.1.b) presents the block diagram of tree

term controller .

Figure (4.1.b) General structure of discrete time

PID controller

4.4 Action of Proportional-Integral-Derivative

Controller

pk

)/1( zzTTk dp

GAActuator

Gp Process

+- ++)(tu

29

Chapter 4

Proportional: To handle the immediate error, the

error is multiplied by a constant P (for

proportional), and added to the controlled quantity.

Integral: To learn from the past, the error is

integrated (added up) over a period of time, and

then multiplied by a constant (making an average),

and added to the controlled quantity. A simple

proportional system either oscillates, moving back

and forth around the set point because there’s

nothing to remove the error when it overshoots, or

oscillates and/or stabilizes at a too low or too

high value. By adding a proportion of the average

error to the process input, the average difference

between the process output and the set point is

continually reduced.

Derivative: To handle the future, the first

derivative (the slope of the error) over time is

calculated, and multiplied by another constant D,

and also added to the controlled quantity. The

derivative term controls the response to a change in

the system. The larger the derivative term, the more

rapidly the controller responds to changes in the

process’s output. In table (4.1) below we can see

30

Chapter 4

the effect of Independent P, I and D closed

response.

Effect of independent P,I, and D tuning on closed

loop responseIncreasin

g

Rise

time

Over

shoot

Settling

time

Steady state

errorKp Decrease Increase Increase DecreaseKi Decrease Increase Increase DecreaseKd Decrease Decrease Decrease Increase

Table (4.1) Effect of Independent P, D and I closed

response

4.5 Assessment of the Ziegler- Nichols, (Z-N)

Methods

The Ziegler-Nichols tuning rules were developed to

give closed loop systems with good attenuation of

load disturbances. The methods were based on

extensive simulations. The design criterion was

quarter amplitude decay ratio, which means that the

amplitude of an oscillation should be reduced by a

factor of four over a whole period. This corresponds

to closed loop poles with a relative damping, which

is too small. Controllers designed by the Ziegler-

31

Chapter 4

Nichols rules thus inherently give closed loop

systems with poor robustness. It also turns out that

it is not sufficient to characterize process

dynamics by two parameters only. The methods

developed by Ziegler and Nichols have been very

popular in spite of these drawbacks. Practically all

manufacturers of controller have used the rules with

some modifications in recommendations for controller

tuning.

One reason for the popularity of the rules is that

they are simple and easy to understand. The tuning

rules give ball park figures. Final tuning is then

done by trial and error. Another bad reason is that

the rules lend themselves very well to simple

exercises for control education. With the insight

into controller design that has developed over the

years it is possible to develop improved tuning

rules that are almost as simple as the Ziegler-

Nichols rules. These rules are developed by starting

with a solid design method that gives robust

controllers with effective disturbance attenuation.

We illustrate with some rules where the process is

characterized by three parameters [12]. Ku and Pcr

32

Chapter 4

are obtained by experiment. Ku is the critical gain

where the shaft exhibits sustained oscillations. Tu

designates the period of these oscillations.

TdTiKpController

0∞0.5kuPProportional

0(1/1.2)T

u0.45kuPI

Proportional

Integral

0.125Tu0.5Tu0.6kuPID

Proportional

Integral

Derivative

Table (4.2) the value of parameters Kp, Ti and Td using

the (Z-N)

4.6 Tuning Techniques

There are three schools of thought on how to select

the values of P, I, and D required achieving an

acceptable level of performance for the controller.

The first method is simple trial and error tweak the

tuning parameters and watches the controller handle

the next error. If it can eliminate the error in a

timely fashion, quit. If it proves to be too

conservative or too aggressive, increase or decrease

one or more of the tuning constants. Experienced

33

Chapter 4

control engineers seem to know just how much

proportional, integral, and derivative action to add

or subtract in order to correct the performance of a

poorly tuned controller.

Unfortunately, intuitive tuning procedures can be

difficult to develop because a change in one tuning

constant tends to affect the performance of all

three terms in the controller's output. For example,

turning down the integral action reduces overshoot.

This in turn slows the rate of change of the error

and thus reduces the derivative action as well.

Therefore, it is desirable to combine the advantages

of both PID and pole-placement which can be easily

tuned. In the next section we will study some PID

pole-placement control design [11].

4.8 PID Pole-Placement Controllers (PID-PPC)

A controller based on the pole placement method in a

closed feedback control loop is designed to

stabilise the closed control loop whilst the

characteristic polynomial should have a previously

determined. Digital PID controllers are possible can

34

Chapter 4

be expressed in the form of a discrete transfer

function (for the derivation see Appendix A1):

(4.8)

The coefficients and are related to

and the proportional, derivation and integral

gain setting by:

(4.9)

(4.10)

(4.11)

In the order to realize our design, we must assume

that the system to be controlled has the following

special structure [13]:

(4.12)

The design process begins by combining the system

model given by equation (4.12) with the controller

of equation (4.8) to obtain the closed loop equation

relating and Thus:

+

35

Chapter 4

(4.13)

Figure (4.2) PID Pole-Placement controller (PID-PPC)

scheme

Here can now select the coefficients and to

give the desired closed loop performance. The aim of

our design is to locate the poles of closed loop

system at their desired positions given by the

following polynomial:

(4.14)

Also can now determine the controller coefficients,

and which give the desired closed characteristic

equation.

(4.15)

210 fff )1(

11 z A

Bz 1

A1

22

110

zfzff

)(tu)(tw )(ty

)(t

++

-+

Controller

36

Chapter 4

By equating coefficients of like power of the

following solution for controller settings is

obtained.

(4.16)

(4.17)

(4.18)

It is obvious from the equation (4.12), (4.13),

(4.14) and (4.15) that in order to achieve PID pole-

placement control presented in section (4.78), the

mode of the process must be described by the same

model given by equation (4.12).

Therefore, the PID Pole-Placement controller

presented in this section suffers from the following

drawback.

1- It can only be applied to limited process models

as described by equation (4.12).

37

Chapter 4

2- If the time delay is more than one, the

controller cannot be used.

In order to overcome the second limitation a Smith

Predictor can be used. In order to overcome both

disturbances as model reduction method can be used

and then the PID Pole-Placement given (4.7) can be

applied.

However, if the parameters obtained from the model

reduction method are not accurate, the output

response will be affected. Therefore a new PID Pole-

Placement controller is developed to overcome all

the limitations mentioned above.

4.8 Smith Predictor Controller (SPC)

In the late 1950s, O. I. M. Smith proposed a

controller that became known as a Smith Predictor. He

first suggested this control scheme for factory

processes with long transport delays, for example

catalytic crackers and steel mills, but the idea can

be generalized to all control processes that have

long loop delays[14].

4.9 Smith Predictor Controller Idea:

38

Chapter 4

The presence of large delays reduces the achievable

control performance [15]. In all real life system, k

will at least be one, because all meaning full

systems take a none zero amount of time to respond

to external stimuli. If there is a transport delay

in implementing the control effort, k will be larger

than one. Such a situation arises also when the

plants are inherently sluggish and they take some

time respond to control efforts. Chemical processes

they have this short coming. In chemical engineering

terminology, the time to respond to external inputs

is known as dead time. In all these cases k could be

a large number. The presence of a large delay k

implies that the control action will be delayed by

the same extent and the large delay, the worse the

control performance will be. Because of the adverse

effect of long delays in the plant, and would like

to account for them.

In order to carry out the PID design discussed in

section (4.7) the system that includes k >1, the

effect of the any delay more large than one must be

removed, through a strategy known as the smith

39

Chapter 4

predictor. In view of this, will assume that the

plant model given by:

(4.19)

Recall that the number polynomial has the form:

(4.20)

Defining:

(4.21)

Equation (4.19) becomes:

(4.22) and have defined that has one delay,

the minimum which expected in real applications. Now

looking for ways to get rid of the adverse effects

of the delay term Towards this end, consider

the following equation:

(4.23) Where , and can be thought of as

estimates of and , respectively.

40

Chapter 4

F

igure (4.3) PID structure of Smith predictor

controller

When a good knowledge of the plant, the estimates

has becomes exact, and equation (4.23) becomes:

(4.24)

Thereby getting rid of the adverse effects of

in equation (4.22). This can be treated as the

equivalent model of the original plant given by

equation (4.19). Figure (4.3) shows structure of

this idea, where have discussed, and PID pole-

placement controller in section (4.7). Therefore, if

the good knowledge of plant as a result, the model

parameters will be identical to those of original

0H1

ABz dk 1

mAB

dm)1(1 mkz

22

110

zfzff

+-_

)(t

++

)(tw )(tu )(ty

pyy ˆPID

41

Chapter 4

plant model. That is, , and . Then

can be measured as:

(4.25)

In the figure (4.4) can see the effect of smith

predictor on paper machine printer (case study 4).

The discrete time model of this system given by

[16]:

, where the parameters are ,

, ,

, With time delay (k=3).

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t

)(tyAfter SPCBefore SPC

42

Chapter 4

Figure (4.4): Effect of smith predictor on paper

machine printer

Hence, the identical PID pole-placement control

design steps can be followed and the effect of the

extra delays is removed. It is clear from equation

(4.7) that Smith predictor reduces the time delay to

one. This allows us to use PID pole-placement

discussed in section (4.7). However, this

modification is not enough in the situation when the

polynomial is of order more than zero. For

such process a model reduction method or New

Modified PID Pole-Placement Controller (NM-PID-PPC),

which is proposed in section (4.10) and (4.11), can

be used.

4.10 Model Reduction

The capacity of a model to accurately describe a

system seems to increase with the order of the

model, in practice, models with low orders are

required in many situations. In some cases, the

amount of information contained in a complex model

may obfuscate simple (some of our cases), insightful

behaviors, which can be better captured and explored

43

Chapter 4

by a model with low order. In cases such as control

design and filtering, where the design procedures

might be computationally very demanding, limited

computational resources certainly benefit from low

order models. These examples justify the need to

develop procedures that are able to approximate

complex high order models by generating adequate

reduced order models. As a result, some degree of

detailing will be permanently lost in the reduced

order model. The differences between the dynamics of

the high order model and the obtained low order

model (the unmodeled dynamics) can be often taken

into account in the low order model as a noise,

which can be handled using stochastic process

methods. In any case, the model reduction procedures

might be flexible enough to let the user indicate

the essential behaviors that need to be captured for

its application. There are many model reduction

methods can be used. In this thesis, the model

reduction based least squares is used. The least

squares (model reduction method) are situated in the

Appendix (A2). Figure (4.5) shows the block diagram

44

Chapter 4

of the PID Pole-Placement controller based on least

squares model reduction [17].

In the following section (4.11), a more effective

PID Pole-Placement is proposed.

The model is implemented to one case study and the

simulation results are show in appendix (A2).

Figure (4.5) Block diagram of the PID Pole-Placement

controller based on LS model reduction

4.11 New Modified PID Pole-Placement Controller (NM-

PID-PPC)

Here can see the modified pole-placement control

design discussed in section (3.4), is extended to a

PID controller )(

)(11

zAzBz k

A1

22

11

01

1

zazabz

PID Control design

)(te)(tw )(tu

Process

System model (off line identification)

)(t

)(ty+-++

45

Chapter 4

PID controller. The modified pole-placement control

low given by equation (3.14) can by written as [18]:

(4.26)

Let assume that is of order 2, then the

polynomial is used to find the PID controller

parameters. The polynomial acts as a compensator

[6]. Combining equation (4.26) and (3.1) gives the

closed loop system as follows:

(4.27)

Assuming that:

(4.28)

And let:

(4.29)

Where

(4.30)

(4.31)

(4.32)

46

Chapter 4

It can be seen from the equation (4.8), (4.9),

(4.10), (4.11) and (4.30), the PI control is

achieved if the order of the plant is one and a PID

controller is obtained if the plant is second order.

Figure (4.6) New Modified PID Pole-Placement Control

system

If the plant is of order more than two, a PID

controller plus additional compensator is achieved

[18]. Figure (4.6) shows the New modified PID pole-

placement control design.

1

G ~1 SYSETM210

~~~ fff

22

110

~~~ zfzff

NM-PID -PPC

)(ty)(tw

2102

21 2

fffkfk

ffk

I

D

P

Plant

Pole placement

Compensator

-+

+

47